is the reduced temperature of a grain of radius \(a_k\)  and temperature \(T\) , while \(k_\mathrm {B}\)  is the Boltzmann constant. Furthermore, \(f(Z,\tau _k)\)  is the distribution function of grain charges for a grain of reduced temperature \(\tau _k\) , and \(J_\mathrm {i}\) , \(J_\mathrm {s}\) , and \(J_\mathrm {e}\)  are the fluxes of species \(i\) , \(s\) , and electrons onto the grains, respectively. The fluxes are [1]}
\(J_j(Z,\tau _k) = n_j s_j v_j \pi a_k^2 \tilde{J}(Ze/q_j,\tau _k),\) 
